Harish-Chandra bimodules over rational Cherednik algebras

TL;DR

This paper investigates Harish-Chandra bimodules over rational Cherednik algebras, classifies parameter pairs for their existence, and describes their categories, especially for symmetric groups, using localization and symmetry techniques.

Contribution

It provides a classification of Harish-Chandra bimodules over rational Cherednik algebras, including explicit results for symmetric groups, and introduces methods to analyze their structure.

Findings

Classified parameter pairs for fully supported bimodules.

Described the category of bimodules modulo those without full support.

Classified all irreducible bimodules for symmetric groups.

Abstract

We study Harish-Chandra bimodules over the rational Cherednik algebra $H_{c}(W)$ associated to a complex reflection group $W$ with parameter $c$. Our results allow us to partially reduce the study of these bimodules to smaller algebras. We classify those pairs of parameters $(c,c')$ for which there exist fully supported Harish-Chandra bimodules, and give a description of the category of all Harish-Chandra bimodules modulo those without full support. When $W$ is a symmetric group we are able to classify all irreducible Harish-Chandra bimodules. Our proofs are based on localization techniques, the action of the Namikawa-Weyl group on the set of parameters, and the study of partial KZ functors.